Optimal. Leaf size=276 \[ -\frac{d \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{c+d x}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{3 d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d}{2 a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^2}{2 a^3 d}-\frac{3 d x}{2 a^3 f g n \log (F)}+\frac{c+d x}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]
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Rubi [A] time = 0.563778, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {2185, 2184, 2190, 2279, 2391, 2191, 2282, 266, 36, 29, 31, 44} \[ -\frac{d \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{c+d x}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{3 d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d}{2 a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^2}{2 a^3 d}-\frac{3 d x}{2 a^3 f g n \log (F)}+\frac{c+d x}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 2185
Rule 2184
Rule 2190
Rule 2279
Rule 2391
Rule 2191
Rule 2282
Rule 266
Rule 36
Rule 29
Rule 31
Rule 44
Rubi steps
\begin{align*} \int \frac{c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx &=\frac{\int \frac{c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a}-\frac{b \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx}{a}\\ &=\frac{c+d x}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{\int \frac{c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2}-\frac{b \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a^2}-\frac{d \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{2 a f g n \log (F)}\\ &=\frac{(c+d x)^2}{2 a^3 d}+\frac{c+d x}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{c+d x}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{b \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^3}-\frac{d \operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^n\right )^2} \, dx,x,F^{g (e+f x)}\right )}{2 a f^2 g^2 n \log ^2(F)}-\frac{d \int \frac{1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f g n \log (F)}\\ &=\frac{(c+d x)^2}{2 a^3 d}+\frac{c+d x}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{c+d x}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{(c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{2 a f^2 g^2 n^2 \log ^2(F)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^n\right )} \, dx,x,F^{g (e+f x)}\right )}{a^2 f^2 g^2 n \log ^2(F)}+\frac{d \int \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f g n \log (F)}\\ &=\frac{(c+d x)^2}{2 a^3 d}+\frac{c+d x}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{c+d x}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{(c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac{d \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{d \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{2 a f^2 g^2 n^2 \log ^2(F)}\\ &=\frac{(c+d x)^2}{2 a^3 d}-\frac{d}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac{d x}{2 a^3 f g n \log (F)}+\frac{c+d x}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{c+d x}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac{d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}-\frac{d \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac{(c+d x)^2}{2 a^3 d}-\frac{d}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac{3 d x}{2 a^3 f g n \log (F)}+\frac{c+d x}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{c+d x}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac{3 d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}-\frac{d \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}\\ \end{align*}
Mathematica [F] time = 0.938007, size = 0, normalized size = 0. \[ \int \frac{c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 0.056, size = 708, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, d{\left (\frac{3 \, a f g n x \log \left (F\right ) +{\left (2 \,{\left (F^{e g}\right )}^{n} b f g n x \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b\right )}{\left (F^{f g x}\right )}^{n} - a}{2 \,{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} +{\left (F^{f g x}\right )}^{2 \, n}{\left (F^{e g}\right )}^{2 \, n} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + 2 \, \int \frac{2 \, f g n x \log \left (F\right ) - 3}{2 \,{\left ({\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{2} b f g n \log \left (F\right ) + a^{3} f g n \log \left (F\right )\right )}}\,{d x}\right )} + \frac{1}{2} \, c{\left (\frac{2 \,{\left (F^{f g x + e g}\right )}^{n} b + 3 \, a}{{\left (2 \,{\left (F^{f g x + e g}\right )}^{n} a^{3} b n +{\left (F^{f g x + e g}\right )}^{2 \, n} a^{2} b^{2} n + a^{4} n\right )} f g \log \left (F\right )} + \frac{2 \, \log \left (F^{f g x + e g}\right )}{a^{3} f g \log \left (F\right )} - \frac{2 \, \log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{3} f g n \log \left (F\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70884, size = 1565, normalized size = 5.67 \begin{align*} -\frac{3 \,{\left (a^{2} d e - a^{2} c f\right )} g n \log \left (F\right ) + a^{2} d -{\left (a^{2} d f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} c f^{2} g^{2} n^{2} x -{\left (a^{2} d e^{2} - 2 \, a^{2} c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} -{\left ({\left (b^{2} d f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c f^{2} g^{2} n^{2} x -{\left (b^{2} d e^{2} - 2 \, b^{2} c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 3 \,{\left (b^{2} d f g n x + b^{2} d e g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} +{\left (a b d - 2 \,{\left (a b d f^{2} g^{2} n^{2} x^{2} + 2 \, a b c f^{2} g^{2} n^{2} x -{\left (a b d e^{2} - 2 \, a b c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} + 2 \,{\left (2 \, a b d f g n x +{\left (3 \, a b d e - a b c f\right )} g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n} + 2 \,{\left (2 \, F^{f g n x + e g n} a b d + F^{2 \, f g n x + 2 \, e g n} b^{2} d + a^{2} d\right )}{\rm Li}_2\left (-\frac{F^{f g n x + e g n} b + a}{a} + 1\right ) -{\left (2 \,{\left (a^{2} d e - a^{2} c f\right )} g n \log \left (F\right ) + 3 \, a^{2} d +{\left (2 \,{\left (b^{2} d e - b^{2} c f\right )} g n \log \left (F\right ) + 3 \, b^{2} d\right )} F^{2 \, f g n x + 2 \, e g n} + 2 \,{\left (2 \,{\left (a b d e - a b c f\right )} g n \log \left (F\right ) + 3 \, a b d\right )} F^{f g n x + e g n}\right )} \log \left (F^{f g n x + e g n} b + a\right ) + 2 \,{\left ({\left (b^{2} d f g n x + b^{2} d e g n\right )} F^{2 \, f g n x + 2 \, e g n} \log \left (F\right ) + 2 \,{\left (a b d f g n x + a b d e g n\right )} F^{f g n x + e g n} \log \left (F\right ) +{\left (a^{2} d f g n x + a^{2} d e g n\right )} \log \left (F\right )\right )} \log \left (\frac{F^{f g n x + e g n} b + a}{a}\right )}{2 \,{\left (2 \, F^{f g n x + e g n} a^{4} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + F^{2 \, f g n x + 2 \, e g n} a^{3} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{5} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 a c f g n \log{\left (F \right )} + 3 a d f g n x \log{\left (F \right )} - a d + \left (2 b c f g n \log{\left (F \right )} + 2 b d f g n x \log{\left (F \right )} - b d\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{2 a^{4} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} + 4 a^{3} b f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )}^{2} + 2 a^{2} b^{2} f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )}^{2}} + \frac{\int - \frac{3 d}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \frac{2 c f g n \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \frac{2 d f g n x \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx}{2 a^{2} f g n \log{\left (F \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x + c}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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